The Reds have already swooped for the Saints’ captain and top scorer from last term in Adam Lallana and Rickie Lambert and are also eyeing £20m centre-back Dejan Lovren.
And now Brendan Rodgers also wants Rodriguez, who impressed before a serious knee injury ended his campaign and World Cup hopes.
J-Rod is not due to return to action until October, but reports on his fitness have been encouraging.
This has tempted Rodgers, who is keen to re-invest the £75m fee received for Luis Suarez in his first-team in a bid to cope with the rigours of Premier League and Champions League action next season.
The 24-year-old linked well with Lambert and Lallana to net 15 goals in all competitions during 2013/14, which impressed the Anfield chief.
As well as Rodriguez, Liverpool are chasing Swansea’s Wilfried Bony and QPR striker Loic Remy.
Spanish flop Iago Aspas may be offloaded as part of a deal to snap up £18m-rated Sevilla left-back Alberto Moreno in what will be a busy summer of transfers on Merseyside.